etc. Here X i(tθ) denotes the [(T - 2) × K] matrix of x levels obtained by deleting
rows t and θ from Xi., and ∆Xi(t) denotes the [(T - 2) × K] matrix of x differences
obtained by stacking all one-period differences between rows of Xi. not including
period t and the single two-period difference between the columns for periods t +1
and t — 1. The vectors yi(tθ) and ∆yi(t) are constructed from yi∙ in a similar way.
In general, we let subscripts (tθ) and (t) on a matrix or vector denote deletion of
(tθ) differences and t levels, respectively. Stacking yi0(tθ) , ∆yi0(t) , xi(tθ) , and ∆xi(t)
by individuals, we get
y1( tθ ) |
δ y01( t ) |
x1(tθ) |
δ x 1( t ) | ||||
Y(tθ)= |
. . . |
, δ Y ( t ) = |
. . . |
, x ( tθ ) = |
. . . |
, ∆ X ( t ) = |
. . . |
y0N (tθ) |
δ yN ( t ) |
xN (tθ) |
∆xN (t) |
which have dimensions (N × (T — 2)), (N × (T — 2)), (N × (T — 2)K), and (N ×
(T — 2)K), respectively. These four matrices contain the alternative IV sets in the
GMM procedures to be considered below.
Equation in differences, IV’s in levels. Using X (tθ) as IV matrix for ∆Xtθ ,we
obtain the following estimator of β, specific to period (t, θ) differences and utilizing
all admissible x level IV’s,
(38)
βDx(tθ) = (∆Xtθ) 0X (tθ) X (0tθ)X (tθ)
1 X(0tθ) (∆Xtθ)
-1
(∆Xtθ) 0X (tθ) X (0tθ) X (tθ)
X (0tθ) (∆ytθ )
= hPi(∆xitθ) 0x
i(tθ)ihPi xi0(tθ) xi(tθ) i hPi xi0(tθ) (∆xitθ
-1
hPi (∆xitθ) xi(tθ) ihPi xi(tθ) xi(tθ) i hPi xi
It exists if X (0tθ)X (tθ) has rank (T — 2)K, which requires N ≥ (T — 2)K. This
estimator examplifies (23), utilizes the OC E[x0(tθ)(∆eitθ)] = 0(T-2)K, 1 - which
follows from (32) - and minimizes the quadratic form
[N-1X(0tθ)∆tθ]0[N-2X(0tθ)X(tθ)]-1[N-1X(0tθ)∆tθ].
18